# Gravitational dynamics in a 2+1+1 decomposed spacetime along nonorthogonal double foliations: Hamiltonian evolution and gauge fixing

## TL;DR

This paper introduces a novel 2+1+1 spacetime decomposition using nonorthogonal double foliations, enabling Hamiltonian analysis and gauge fixing in complex gravitational scenarios with both temporal and spatial symmetries.

## Contribution

It generalizes previous orthogonal foliations by incorporating a new metric variable related to Lorentz rotations and vorticity, facilitating Hamiltonian dynamics and gauge fixing in scalar-tensor theories.

## Key findings

- Developed a new 2+1+1 decomposition formalism.
- Derived Hamiltonian equations for general relativity.
- Established gauge-fixing for scalar perturbations in spherically symmetric spacetimes.

## Abstract

Motivated by situations with temporal evolution and spatial symmetries both singled out, we develop a new 2+1+1 decomposition of spacetime, based on a nonorthogonal double foliation. Time evolution proceeds along the leaves of the spatial foliation. We identify the gravitational variables in the velocity phase-space as the 2-metric (induced on the intersection $\Sigma_{t\chi }$ of the hypersurfaces of the foliations), the 2+1 components of the spatial shift vector, together with the extrinsic curvature, normal fundamental form and normal fundamental scalar of $\Sigma _{t\chi }$, all constructed with the normal to the temporal foliation. This work generalizes a previous decomposition based on orthogonal foliations, a formalism lacking one metric variable, now reintroduced. The new metric variable is related to (i) the angle of a Lorentz-rotation between the nonorthogonal bases adapted to the foliations, and (ii) to the vorticity of these basis vectors. As a first application of the formalism, we work out the Hamiltonian dynamics of general relativity in terms of the variables identified as canonical, generalizing previous work. As a second application we present the unambiguous gauge-fixing suitable to discuss the even sector scalar-type perturbations of spherically symmetric and static spacetimes in generic scalar-tensor gravitational theories, which has been obstructed in the formalism of orthogonal double foliation.